Stochastic Solver Research Articles

Toward a Full Configurational Accuracy Calculation of an Arbitrary Molecule via Fragment Embedding and a Stochastic Solver.

We demonstrate the feasibility of using a stochastic solver, full configuration interaction quantum Monte Carlo with the initiator approximation (i-FCIQMC), to converge fragment embedding calculations, namely bootstrap embedding (BE). We first propose and test a general protocol for converging BE-i-FCIQMC calculations and then suggest how the quality of the calculation compares against that of deterministic BE-FCI using different numbers of walkers. We then demonstrate that BE-i-FCIQMC can perform as well as BE-FCI in the large walker limit and how different factors, including the size of the Hilbert space of the fragments, the number of walkers, and the nature of the chemical system, affect the achievable matching error. We finally perform BE-FCI calculations in realistic systems like benzene and cyclohexane using a double-ζ basis set. This work demonstrates the potential of performing FCI quality calculations in realistic systems using BE.

Datacentric Similarity Matching of Emergent Stigmergic Clustering to Fractional Factorial Vectoring: A Case for Leaner-and-Greener Wastewater Recycling

Water scarcity is a challenging global risk. Urban wastewater treatment technologies, which utilize processes based on single-stage ultrafiltration (UF) or nanofiltration (NF), have the potential to offer lean-and-green cost-effective solutions. Robustifying the effectiveness of water treatment is a complex multidimensional characteristic problem. In this study, a non-linear Taguchi-type orthogonal-array (OA) sampler is enriched with an emergent stigmergic clustering procedure to conduct the screening/optimization of multiple UF/NF aquametric performance metrics. The stochastic solver employs the Databionic swarm intelligence routine to classify the resulting multi-response dataset. Next, a cluster separation measure, the Davies–Bouldin index, is used to evaluate input and output relationships. The self-organized bionic-classifier data-partition appropriateness is matched for signatures between the emergent stigmergic clustering memberships and the OA factorial vector sequences. To illustrate the proposed methodology, recently-published multi-response multifactorial L9(34) OA-planned experiments from two interesting UF-/NF-membrane processes are examined. In the study, seven UF-membrane process characteristics and six NF-membrane process characteristics are tested (1) in relationship to four controlling factors and (2) to synchronously evaluate individual factorial curvatures. The results are compared with other ordinary clustering methods and their performances are discussed. The unsupervised robust bionic prediction reveals that the permeate flux influences both the UF-/NF-membrane process performances. For the UF process and a three-cluster model, the Davies–Bouldin index was minimized at values of 1.89 and 1.27 for the centroid and medoid centrotypes, respectively. For the NF process and a two-cluster model, the Davies–Bouldin index was minimized for both centrotypes at values close to 0.4, which was fairly close to the self-validation value. The advantage of this proposed data-centric engineering scheme relies on its emergent and self-organized clustering capability, which retraces its appropriateness to the fractional factorial rigid structure and, hence, it may become useful for screening and optimizing small-data wastewater operating conditions.

Entropy optimization of non-Newtonian hybrid nanofluid EMHD flow by numerical and Levenberg–Marquardt backpropagation approach over a rotating disk

The goal of this study is to analyze the heat transfer characteristics of a Casson hybrid nanofluid flow system by incorporating EMHD (electromagnetohydrodynamic) and entropy production using the stochastic solver and Levenberg-Marquardt backpropagation neural networks. Self-similarity transformations are used to turn the mathematical model of a system of partial into a series of ordinary differential equations. The datasets are built through Runge–Kutta fourth order with a shooting approach, which aids in the generation of a continuous neural network mapping. To grasp the neural network mapping, the validation, training and testing processes are used to estimate the solutions of various physical constraints. For this hybrid model, the determination, convergence, verification and stability of Levenberg-Marquardt backpropagation neural network mappings are evaluated using regression-based statistical analysis, mean squared error and error histograms. To acquire high possibilities in drug delivery systems, hybrid nanofluids are regarded as blood-based nanofluids with dispersion of hybrid ( and ) nanoparticles. Due to this reason, this model might be beneficial for medication delivery systems, particularly if pharmaceuticals are injected into the bloodstream.

Sensitivity of a homogeneous and isotropic second‐gradient continuum model for particle‐based materials with respect to uncertainties

AbstractThis work concerns the probabilistic analysis of particle‐based materials. More precisely, this work is devoted to the stochastic modeling of the geometric and constitutive microscale parameters associated with particle‐pair interactions of an existing model for particle‐based materials. Such an issue is addressed with a probabilistic methodology that relies on the maximum entropy principle from information theory. After defining and improving the chosen second‐gradient continuum model for particle‐based materials, it is shown that for micro‐homogeneous and micro‐isotropic materials, the involved microscale parameters turn out to be statistically independent. More precisely, the particle‐pair distance between two consecutive particles is a uniformly distributed random variable and the specific micro‐scale stiffness parameters are Gamma‐distributed random variables. A homogeneous and isotropic 2D concrete plate subjected to an axial load is considered for illustration purposes. A stochastic solver based on Monte Carlo numerical simulations and mixed finite element (FE) method are chosen. The FE discretization is applied to the weak formulation of the equivalent continuum model with random mechanical properties. On the contrary, the particle‐pair distance and the micro‐scale stiffness parameters, which are the parameters of the boundary value problem formulated for the equivalent continuum model, are modeled as random variables. Finally, uncertainties propagation is discussed and statistical fluctuations of the macro mechanical response are found to be significant.

RETRACTED ARTICLE: A fractional order nonlinearmodel of the love story of Layla and Majnun

In this study, a fractional order mathematical model using the romanticrelations of the Layla and Majnun is numerically simulated by the Levenberg–Marquardtbackpropagation neural networks. The fractional order derivatives provide more realisticsolutions as compared to integer order derivatives of the mathematical model based onthe romantic relationship of the Layla and Majnun. The mathematical formulation of thismodel has four categories that are based on the system of nonlinear equations. Theexactness of the stochastic scheme is observed for solving the romantic mathematicalsystem using the comparison of attained and Adam results. The data for testing,authorization, and training is provided as 15%, 75% and 10%, along with the twelvenumbers of hidden neurons. Furthermore, the reducible value of the absolute errorimproves the accuracy of the designed stochastic solver. To prove the reliability ofscheme, the numerical measures are presented using correlations, error histograms, statetransitions, and regression.

QUANTIFICATION AND PROPAGATION OF MODEL-FORM UNCERTAINTIES IN RANS TURBULENCE MODELING VIA INTRUSIVE POLYNOMIAL CHAOS

Undeterred by its inherent limitations, Reynolds-averaged Navier-Stokes (RANS) based modeling is still considered the most recognized approach for several computational fluid dynamics (CFD) applications. Recently, in the turbulence modeling community, quantification of model-form uncertainties in RANS has attracted significant interest. We present a stochastic RANS solver with an efficient implementation of the intrusive polynomial chaos (IPC) method in OpenFOAM. The stochastic solver quantifies and propagates the uncertainties associated with the output of the RANS model (eddy viscosity or Reynolds stress tensor). Two distinct high-dimensional variants of the uncertainties are considered, namely, the random eddy viscosity field (REVF) and the random Reynolds stress tensor field (RRSTF). The randomness is introduced in the approximated eddy viscosity field and the Reynolds stress tensor, while asserting the realizability. The stochastic RANS solver has been tested on various benchmark problems for RANS turbulence modeling. In this study, we discuss two important problems where the stochastic RANS solver shows significantly better performance than the traditional uncertainty quantification (UQ) methods. The first problem analyzed is the flow over periodic hills with a REVF, while the second stochastic problem considered is the flow in a square duct with a RRSTF. Along with the comparison for three different RANS turbulence models, a detailed analysis of the stochastic solver based on various influential model parameters is also presented. The IPC based stochastic solver demonstrated the potential to be used in the UQ analysis of further complex CFD applications, especially when a large number of deterministic simulations is not feasible, e.g., wind farm CFD simulations.

COMPUTATIONAL PERFORMANCES OF MORLET WAVELET NEURAL NETWORK FOR SOLVING A NONLINEAR DYNAMIC BASED ON THE MATHEMATICAL MODEL OF THE AFFECTION OF LAYLA AND MAJNUN

The aim of this study is to design a novel stochastic solver through the Morlet wavelet neural networks (MWNNs) for solving the mathematical Layla and Majnun (LM) system. The numerical representations of the mathematical LM system have been presented by using the MWNNs along with the optimization is performed through the hybridization of the global and local search schemes. The local active-set (AS) and global genetic algorithm (GA) operators have been used to optimize an error-based merit function using the differential LM model and its initial conditions. The correctness of the MWNNs using the local and global operators is observed through the comparison of the obtained solutions and the Adams scheme, which is used as a reference solution. For the stability of the MWNNs using the global and local operators, the statistical performances with different operators have been provided using the multiple executions to solve the nonlinear LM system.

A mathematical model of coronavirus transmission by using the heuristic computing neural networks

In this study, the nonlinear mathematical model of COVID-19 is investigated by stochastic solver using the scaled conjugate gradient neural networks (SCGNNs). The nonlinear mathematical model of COVID-19 is represented by coupled system of ordinary differential equations and is studied for three different cases of initial conditions with suitable parametric values. This model is studied subject to seven class of human population N(t) and individuals are categorized as: susceptible S(t), exposed E(t), quarantined Q(t), asymptotically diseased IA(t), symptomatic diseased IS(t) and finally the persons removed from COVID-19 and are denoted by R(t). The stochastic numerical computing SCGNNs approach will be used to examine the numerical performance of nonlinear mathematical model of COVID-19. The stochastic SCGNNs approach is based on three factors by using procedure of verification, sample statistics, testing and training. For this purpose, large portion of data is considered, i.e., 70%, 16%, 14% for training, testing and validation, respectively. The efficiency, reliability and authenticity of stochastic numerical SCGNNs approach are analysed graphically in terms of error histograms, mean square error, correlation, regression and finally further endorsed by graphical illustrations for absolute errors in the range of 10−05 to 10−07 for each scenario of the system model.

NUMERICAL COMPUTING TO SOLVE THE NONLINEAR CORNEAL SYSTEM OF EYE SURGERY USING THE CAPABILITY OF MORLET WAVELET ARTIFICIAL NEURAL NETWORKS

In this study, a novel heuristic computing technique is presented to solve bioinformatics problem for the corneal shape model of eye surgery using Morlet wavelet artificial neural network optimized by the global search schemes, i.e. genetic algorithm (GA), local search technique, i.e. sequential quadratic programming (SQP) and the hybrid of GA-SQP. To measure the performance of the design network configuration, different cases based on nonlinear second-order differential equations governing the corneal model have been solved effectively. The numerical procedure of Adams method is implemented for the comparison purpose of the presented outcomes of the stochastic solver, which shows the worth of the present scheme based on accuracy and convergence with negligible values of absolute error in the range 10[Formula: see text] to 10[Formula: see text]. Furthermore, statistical measures are presented based on “mean absolute error”, “root mean square error” and “coefficient of Theil’s inequality” which additionally endorsed consistently accurate performance of integrated intelligent computing framework for solving the corneal shape model.

Design of neuro-swarming computational solver for the fractional Bagley\u2013Torvik mathematical model

This study is to introduce a novel design and implementation of a neuro-swarming computational numerical procedure for numerical treatment of the fractional Bagley–Torvik mathematical model (FBTMM). The optimization procedures based on the global search with particle swarm optimization (PSO) and local search via active-set approach (ASA), while Mayer wavelet kernel-based activation function used in neural network (MWNNs) modeling, i.e., MWNN-PSOASA, to solve the FBTMM. The efficiency of the proposed stochastic solver MWNN-GAASA is utilized to solve three different variants based on the fractional order of the FBTMM. For the meticulousness of the stochastic solver MWNN-PSOASA, the obtained and exact solutions are compared for each variant of the FBTMM with reasonable accuracy. For the reliability of the stochastic solver MWNN-PSOASA, the statistical investigations are provided based on the stability, robustness, accuracy and convergence metrics.

Design of evolutionary finite difference solver for numerical treatment of computer virus propagation with countermeasures model

In the present study, a novel application of integrated evolutionary computing paradigm is presented for the analysis of nonlinear systems of differential equations representing the dynamics of virus propagation model in computer networks by exploiting the discretization strength of finite difference procedure, global search efficacy of genetic algorithms (GAs) aided with interior-point method (IPM) as efficient local search mechanism. Residual error based cost function is constructed by utilizing the effectiveness of approximation in mean square error sense and combined strength of GA-IPM is used as a viable optimization mechanism to find the solution of the problem. The proposed scheme is implemented for dynamical analysis of the model in terms of susceptible–infected and protected computer nodes by varying the probabilities of infections, countermeasures, curing and immunity while keeping connected and disconnected rates fixed. The statistical performance evaluated by means of deviations from reference Adams numerical results are practiced viably through performance metrics of accuracy and computational complexity to demonstrate the worth of integrated stochastic solver.

Competency of Neural Networks for the Numerical Treatment of Nonlinear Host-Vector-Predator Model.

The aim of this work is to introduce a stochastic solver based on the Levenberg-Marquardt backpropagation neural networks (LMBNNs) for the nonlinear host-vector-predator model. The nonlinear host-vector-predator model is dependent upon five classes, susceptible/infected populations of host plant, susceptible/infected vectors population, and population of predator. The numerical performances through the LMBNN solver are observed for three different types of the nonlinear host-vector-predator model using the authentication, testing, sample data, and training. The proportions of these data are chosen as a larger part, i.e., 80% for training and 10% for validation and testing, respectively. The nonlinear host-vector-predator model is numerically treated through the LMBNNs, and comparative investigations have been performed using the reference solutions. The obtained results of the model are presented using the LMBNNs to reduce the mean square error (MSE). For the competence, exactness, consistency, and efficacy of the LMBNNs, the numerical results using the proportional measures through the MSE, error histograms (EHs), and regression/correlation are performed.

A stochastic solver based on the residence time algorithm for crystal plasticity models

The deformation of crystalline materials by dislocation motion takes place in discrete amounts determined by the Burgers vector. Dislocations may move individually or in bundles, potentially giving rise to intermittent slip. This confers plastic deformation with a certain degree of variability that can be interpreted as being caused by stochastic fluctuations in dislocation behavior. However, crystal plasticity (CP) models are almost always formulated in a continuum sense, assuming that fluctuations average out over large material volumes and/or cancel out due to multi-slip contributions. Nevertheless, plastic fluctuations are known to be important in confined volumes at or below the micron scale, at high temperatures, and under low strain rate/stress deformation conditions. Here, we develop a stochastic solver for CP models based on the residence-time algorithm that naturally captures plastic fluctuations by sampling among the set of active slip systems in the crystal. The method solves the evolution equations of explicit CP formulations, which are recast as stochastic ordinary differential equations and integrated discretely in time. The stochastic CP model is numerically stable by design and naturally breaks the symmetry of plastic slip by sampling among the active plastic shear rates with the correct probability. This can lead to phenomena such as intermittent slip or plastic localization without adding external symmetry-breaking operations to the model. The method is applied to body-centered cubic tungsten single crystals under a variety of temperatures, loading orientations, and imposed strain rates.

Neuro-intelligent mappings of hybrid hydro-nanofluid Al2O3–Cu–H2O model in porous medium over rotating disk with viscous dissolution and Joule heating

The aim of study is to investigate the mass and heat transfer phenomena in hybrid hydro-nanofluidic system involving Al2O3–Cu–H2O over the rotating disk in porous medium with viscous dissolution and Joule heating through the stochastic solver by way of Levenberg-Marquardt backpropagation neural networks. The mathematical model in system of PDEs describes the physical phenomena of the hybrid hydro-nanofluid flow problem are converted into set of ODEs by means of scaling group transformations. The datasets are constructed by utilizing the power of explicit Runge-Kutta numerical method that help to the develop a continuous neural networks mapping. The validation, training and testing processes are utilized to learn the neural network mapping to estimate the solution of various scenarios with cases that are constructed by varying different values of physical constraints such as porosity factor, inertia coefficient, Prandtl number, Brinkman number, radiation parameter, mgnetic parameter, concentration of nanoparticles on the velocities and temperature profiles. Determination, convergence, verification and stability of Levenberg-Marquardt backpropogation neural network mappings are validated on the assessment of achieved accuracy through regression based statistical analysis, mean squared error and error histograms for hybrid hydro-nanofluidic model.

Deep learning-based least squares forward-backward stochastic differential equation solver for high-dimensional derivative pricing

We propose a new forward-backward stochastic differential equation solver for high-dimensional derivative pricing problems by combining a deep learning solver with a least squares regression technique widely used in the least squares Monte Carlo method for the valuation of American options. Our numerical experiments demonstrate the accuracy of our least squares backward deep neural network solver and its capability to produce accurate prices for complex early exercisable derivatives, such as callable yield notes. Our method can serve as a generic numerical solver for pricing derivatives across various asset groups, in particular, as an accurate means for pricing high-dimensional derivatives with early exercise features.

Intelligent networks for crosswise stream nanofluidic model with Cu–H2O over porous stretching medium

The porous media transport theories are thoroughly operative to analyse transferral phenomenon in reducing the bio-convective flow instabilities and biological tissues. The present study is designed to investigate the heat transfer phenomena in nanofluidic system involving Cu − H2O over the stretched porous media with the strength of stochastic solver via Levenberg-Marquardt backpropagation networks. The mathematical model of physical phenomena is described in PDEs that are reduced to system of ODEs through scaling group transformations. The datasets are determined through explicit Runge-Kutta numerical method and used as a target parameter for the development of continuous neural networks mapping. The training, testing and validation processes are utilized in learning of neural network models based on backpropagation of Levenberg-Marquardt technique to determines the solution of different scenarios constructed on the various values of porosity parameter along with six different cases based on the stretching ratio values. Validation and verification of neural network model to find the solution of nanfluidic problem is endorsed on the assessment of achieved accuracy through mean squared error, error histograms and regression studies.

Quantification and Analysis of Uncertainties in Parameters of Rotors Supported by Bearings

Rotating unbalance is one of the most important critical parameters that causes operational failures of rotating machinery. The uneven distribution of mass on the structure of the rotors creates heavy spots, which must be eliminated to avoid generating excessive stress on the rotor bearings. The main objective of this work is to perform the uncertainty analysis on rotating machine systems supported with rolling bearings. A computational procedure is implemented to achieve this objective that can qualitatively represent the main behaviors and parameters of rotating machines. Further, methods of uncertainty quantification are applied to verify the behavior of the system given the probability density functions of the input parameters. One of the most commonly used methods is the Monte Carlo method, which requires thousands of simulations to produce accurate results. This method is used to obtain means and standard deviations of the system responses, given the means and standard deviations of the inputs. In our work, Monte Carlo simulation has been successfully used as a reference stochastic solver to evaluating the variability of the dynamic responses.

Quantification and Analysis of Uncertainties in Parameters of Rotors Supported by Bearings

Rotating unbalance is one of the most important critical parameters that causes operational failures of rotating machinery. The uneven distribution of mass on the structure of the rotors creates heavy spots, which must be eliminated to avoid generating excessive stress on the rotor bearings. The main objective of this work is to perform the uncertainty analysis on rotating machine systems supported with rolling bearings. A computational procedure is implemented to achieve this objective that can qualitatively represent the main behaviors and parameters of rotating machines. Further, methods of uncertainty quantification are applied to verify the behavior of the system given the probability density functions of the input parameters. One of the most commonly used methods is the Monte Carlo method, which requires thousands of simulations to produce accurate results. This method is used to obtain means and standard deviations of the system responses, given the means and standard deviations of the inputs. In our work, Monte Carlo simulation has been successfully used as a reference stochastic solver to evaluating the variability of the dynamic responses.

ANALYSIS AND COMPARISON OF APOLLO3® AND TRIPOLI-4® NEUTRON NOISE SOLVERS

Neutron noise analysis addresses the description of small time-dependent flux fluctuations in reactor cores, induced by small global or local perturbations of the macroscopic cross sections due to density fluctuations of the coolant, to vibrations of fuel elements, control rods, or structural materials. The general noise equations are obtained by assuming small perturbations around a steady-state neutron flux and by subsequently taking the Fourier transform in the frequency domain. Recently, new neutron noise solvers have been implemented in diffusion and transport theory in APOLLO3®, the multi-purpose deterministic transport code developed at CEA, and a new stochastic solver has been implemented for the neutron noise analysis in the frequency domain in the Monte Carlo code TRIPOLI-4®, also developed at CEA. In this paper, we compare the two solvers for the case of fuel pin oscillations in a simplified UOX fuel assembly, in view of proposing the examined configurations as a benchmark for neutron noise calculations.

FMNEICS: fractional Meyer neuro-evolution-based intelligent computing solver for doubly singular multi-fractional order Lane–Emden system

In the present study, a novel fractional Meyer neuro-evolution-based intelligent computing solver (FMNEICS) is presented for numerical treatment of doubly singular multi-fractional Lane–Emden system (DSMF-LES) using combined heuristics of Meyer wavelet neural networks (MWNN) optimized with global search efficacy of genetic algorithms (GAs) and sequential quadratic programming (SQP), i.e., MWNN-GASQP. The design of novel FMNEICS for DSMF-LES is presented after derivation from standard Lane–Emden equation, and the singular points and shape factors along with fractional-order terms are analyzed. The MWNN modeling strength is used to represent the system model DSMF-LES in the mean-squared error-based merit function and optimization of the networks is carried out with integrated optimization ability of GASQP. The verification, validation, and perfection of the FMNEICS for three different cases of DSMF-LES are established through comparative studies from reference solutions on convergence, robustness, accuracy, and stability measures. Moreover, the observations through the statistical analysis further authenticate the worth of proposed fractional MWNN-GASQP-based stochastic solver.